No and no. Although the pickups dominate the tone, the guitar's acoustic properties will come through to some extent. I have two guitars with SD Jazz pickups and the same tone and volume controls. One is an OLP bari with 10-46 strings tuned to open D, the other is a 1982 Westone 335 knockoff tuned to open E. Even with a capo on the bari to bring it up to open E, they sound quite different.

If that "les paul" has an Alnico 2, it will bring it close, but not enough to sound like a "les paul". Though that might be due to whole lots different factors than just tonewood like scale length, string gauge, pick gauge... The pickup height needs to be exactly the same on both guitars (of course measured from pickup to string), the player needs to have same attack etc.And then there's Tony Done's post up there.

You can put the same pickups on 2 identical guitars and they still might not sound exactly the same. Unless you are playing them through a Line 6. Those amps don't care what you plug into them.

The high-level DT series will care. POD HDs, while they're not really amps, will care too. You might mean the Spider IV for example, Spider Valve might already let the pickup sound through. Don't just slam a brand just because of one ill-branded and torched product series. POD HD500 is excellent and I'd like to have one at home TBH.